State Standards: GLEs/GSEs National Content Standards:
Grade Span Expectations [Math]: M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems. M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems. M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Grade Span Expectations [Physics]: PS3 (7-8) 8: Students demonstrate and understanding of motion by… (8a)… measuring the distance and time for a given moving object and using those values to calculate speed. (8b)… Solving for any unknown in the expression v=d/t given values for the other two variables. (8c)… Differentiating between speed, velocity and acceleration.
PS3 (9-11) 8: Students demonstrate and understanding of forces and motion by… (8a)… Predicting and/or graphing the path of an object in different reference planes and explain how and why it occurs. (8b)… Using modeling, illustrating, and graphing to explain how distance and velocity change over time for a free falling object.
PS3 (Ext) 8: Students demonstrate an understanding of forces and motion by… (8aa)… Using a quantitative representation of how distance and velocity change over time for a free falling object. (8bb)… Using a quantitative representation of the path of an object that has both horizontal and free fall motion. (8cc)… Modeling, illustrating, graphing, and or quantitatively explaining the path of an object that has both horizontal and free fall motion.
Context of the Lesson Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
This lesson will take place towards the end of the projectile motion unit. It is designed to provide the students with an opportunity to work on problems together that are related to projectile motion. They will be able to apply their knowledge of projectile motion to the various applications that will be presented to them.
Prior to this lesson they should have received instruction on how to break vectors into components and how to treat the motion of an object in the x and y directions independently. They should be proficient with the problem solving steps and should be able to easily identify variables that are given and that need to be solved for. They should also understand how to calculate the range of a projectile from the previous investigations that they completed as a class.
With this prerequisite knowledge the students will be more than ready to successfully complete the problems that will be presented to them in this lesson.
Opportunities to Learn Differentiation: Materials, Learners and Environments
Plans to differentiate instruction: This lesson includes two different segments. During the first segment the students will work together in groups to solve different problems that I will have posted around the room. These problems will have visual representations and some will have models along with them. The second segment involves the students sharing their results with the class and explaining how they solved the problems. These two activities appeal to many different types of learners including those that prefer to work alone and those that want to work cooperatively with others. It also appeals to the visual and kinesthetic learners. In addition to this, throughout the lesson (as I mentioned above) I will have more challenging problems for the advanced students. Each of the problems during the problem session will have a specified difficulty and the advanced students can challenge themselves by attempting to complete the more challenging problems. For the students that may be struggling I will have easier problems that they can start at until they build their self-efficacy and confidence with the material. This will allow them to scaffold nicely to the more difficult problems. I will also walk around the room and provide assistance to these students. I will help them by providing them with hints that go along with each problem. The students can choose to use the hints to help them with the problems until they are able to complete the problems all on their own. This will be further explained in the proceeding sections. Accommodations and modifications: As in the previous lessons the students will be grouped with students of equal ability so that they can work at their own pace during the activity and get the most out of the learning experience. Environment factors: I will position the problems at stations throughout the room. Each problem will be located in an envelope and will be located at either a cluster of desks or a lab bench. Therefore the students (in their groups) can walk around the room to the different stations. I will make sure to clear the paths between the stations so that the flow within the classroom is preserved. Materials: ü Problems relating to projectile motion of varying difficulty. ü Envelopes ü Successive hints (explained below) to go along with each problem.
Objectives
Students will demonstrate their understanding of projectile motion by successfully applying their knowledge to solve several real world problems.
Instructional Procedures
LAUNCH A: When the students enter the room during this instructional period I will hand them each a small index card, which has a given variable, constant, or binary operation written on it. The students will be instructed to form one of the three-kinematics equations. They will be forced to circulate the room and gather students with other variables that could potentially help to form one of the equations. For example if a student is given an index card that has an x on it he can either try to find students to help him make the 1st or the 3rd equation. If he wanted to make the 1st equation (x=xi+vit+1/2at2) he would have to find someone with an xi, vi, a ½, 2 t’s, and a few addition signs. This exercise will help the students to better understand and remember the 3 kinematics equations and can also be an entertaining experience for them. I will not allow them to see their notes to remember the equations they will have to come up with them on their own as a class. I will have passed out just enough symbols to form the 3- kinematics equations that I taught them in lesson 4. EXPLORE A: After this activity I will inform the students that there will be another “Student Challenge” for the day. This lesson will be incredibly similar to the problem session that was held during lesson 2 (1D Kinematics review). I will instruct the students to split up into groups of 2-4 and I will explain the rules of the game that they will play. There will be 15 challenges that will be spread out in envelopes at different stations around the room. Students will have 45-60 minutes to complete as many challenges as they can and record all of their work and their answers on notepaper. Each “challenge” will be color coded in order to designate the difficulty of the task (Black=difficult, blue= intermediate, green=easy) (Yes! I do love to ski!). The more difficult problems are worth more points because they are more in depth and will take more time to complete (Black = 5 points, Blue=3 points, Green=1pt). There will be more “challenges” than there are groups so once a group finishes with a problem they can move to an open station and won’t have to wait for all of the other students to finish with the problem that they are working on. I would also not mind if there are more than one group working on the same problem at once. Note: I will keep small whiteboards at each station so that groups can jot down their thoughts as they try to work through the problems. At the end of the problem session I will collect everyone’s work and I will assign a grade to each student based on the neatness of his or her work. I will be looking to see that they follow the problem solving steps and will not necessarily be looking to see if they got the correct answer. However, for the purpose of the competition I will give groups points for correct answers to problems and I will tally everyone’s points to see which group won the day’s student challenge. Points will not be awarded to groups for partially correct answers so it is more beneficial for students to be thorough and detailed during this activity than for them to speed through all of the problems. The members of the winning group will receive extra credit points towards the final assessment of the unit. If students seem to be struggling with the problems that I have assigned to them I will walk around the classroom and offer them assistance. “What is the question really asking?” “Go back and look at the example that we did together as a class last week and try to follow that same procedure here” “What variables were given and which are we trying to find” “Draw a picture to help yourself visualize the situation” “This particular equation might be useful to you in this situation” “If we know the initial velocity and the angle then we know the initial velocity in both the x and the y direction. How?” Within each envelope (for each problem) I will have three other small sheets of paper. On first sheet I will have drawn a picture of the situation for the students. On the second sheet I will have identified the given variables for the students. And on the third sheet I will have identified the appropriate equations for the problem. I will encourage students not to look at these “hints” but if they need to they can and this should help them to more successfully complete the problems. In the beginning they may need to look at the hints but eventually they should become proficient enough with the problem solving strategies that they can do this on their own. For the students that may find some of these problems too easy I have included several very challenging problems that they can try to complete. This is a good way for them to challenge themselves and test their knowledge of projectile motion. If this is still not enough for them I will give them a practice exam (from the Barron’s AP book) that they can work on. SUMMARIZE/SHARE A: After the students have been given ample time to complete many of the problems I will instruct them that they must all turn in their work so that I can grade it and determine which group won. The students will return to their desks and If there is time remaining in the class period I will ask them if they would like to review any of the problems that they had worked on. We will do some of these problems together as a class and I will have some groups assist me in doing the problems on the whiteboard at the front of the classroom.
Assessment
In order to assess my students during this investigation I will determine whether each student has successfully completed the objective above Launch A: During the launch phase of this lesson I will informally assess the students on their ability to remember and recreate the 3 kinematics equations that are central to solving projectile motion problems. I will observe them as they work together to form the equations out of the index cards that I will have passed out to each of them. Explore A: During the explore phase of this lesson I will walk around the classroom and will informally assess the students on their ability to work cooperatively in groups and on their problem solving skills. I will question the groups while they are working together and try to gauge each students understanding of the material. I will determine if more time may need to be spent on the concepts that had been recently presented to the students. The end of the unit is nearing and I want to make sure that all students will be ready for final assessment that will take place the following week. I may need to ask some students to stay after school to practice more problems with me. Summarize A: The students will pass in their work from the day to me at the end of the problem session. I will grade each individual’s work for completion and will specifically look to make sure that they are explicitly following the steps to problem solving. I want to make sure that they continue to use these steps even when the problems seem overly simple to them. Once they encounter more complicated problems they will need to outline and detail each step in order to be successful. The students will receive a quiz grade for their work and I will not dock points for incorrect answers or if they aren’t able to complete all of the problems.
Reflections This section to be completed only if lesson plan is implemented.
Lesson Implementation: Was not able to implement this lesson yet.
Lesson #8: Student Challenge!
Lesson Title
Student Challenge!
National Content Standards:
M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems.
M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems.
M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Grade Span Expectations [Physics]:
PS3 (7-8) 8: Students demonstrate and understanding of motion by…
(8a)… measuring the distance and time for a given moving object and using those values to calculate speed.
(8b)… Solving for any unknown in the expression v=d/t given values for the other two variables.
(8c)… Differentiating between speed, velocity and acceleration.
PS3 (9-11) 8: Students demonstrate and understanding of forces and motion by…
(8a)… Predicting and/or graphing the path of an object in different reference planes and explain how and why it occurs.
(8b)… Using modeling, illustrating, and graphing to explain how distance and velocity change over time for a free falling object.
PS3 (Ext) 8: Students demonstrate an understanding of forces and motion by…
(8aa)… Using a quantitative representation of how distance and velocity change over time for a free falling object.
(8bb)… Using a quantitative representation of the path of an object that has both horizontal and free fall motion.
(8cc)… Modeling, illustrating, graphing, and or quantitatively explaining the path of an object that has both horizontal and free fall motion.
Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
Prior to this lesson they should have received instruction on how to break vectors into components and how to treat the motion of an object in the x and y directions independently. They should be proficient with the problem solving steps and should be able to easily identify variables that are given and that need to be solved for. They should also understand how to calculate the range of a projectile from the previous investigations that they completed as a class.
With this prerequisite knowledge the students will be more than ready to successfully complete the problems that will be presented to them in this lesson.
Differentiation: Materials, Learners and Environments
This lesson includes two different segments. During the first segment the students will work together in groups to solve different problems that I will have posted around the room. These problems will have visual representations and some will have models along with them. The second segment involves the students sharing their results with the class and explaining how they solved the problems. These two activities appeal to many different types of learners including those that prefer to work alone and those that want to work cooperatively with others. It also appeals to the visual and kinesthetic learners.
In addition to this, throughout the lesson (as I mentioned above) I will have more challenging problems for the advanced students. Each of the problems during the problem session will have a specified difficulty and the advanced students can challenge themselves by attempting to complete the more challenging problems.
For the students that may be struggling I will have easier problems that they can start at until they build their self-efficacy and confidence with the material. This will allow them to scaffold nicely to the more difficult problems. I will also walk around the room and provide assistance to these students. I will help them by providing them with hints that go along with each problem. The students can choose to use the hints to help them with the problems until they are able to complete the problems all on their own. This will be further explained in the proceeding sections.
Accommodations and modifications:
As in the previous lessons the students will be grouped with students of equal ability so that they can work at their own pace during the activity and get the most out of the learning experience.
Environment factors:
I will position the problems at stations throughout the room. Each problem will be located in an envelope and will be located at either a cluster of desks or a lab bench. Therefore the students (in their groups) can walk around the room to the different stations. I will make sure to clear the paths between the stations so that the flow within the classroom is preserved.
Materials:
ü Problems relating to projectile motion of varying difficulty.
ü Envelopes
ü Successive hints (explained below) to go along with each problem.
Instructional Procedures
When the students enter the room during this instructional period I will hand them each a small index card, which has a given variable, constant, or binary operation written on it. The students will be instructed to form one of the three-kinematics equations. They will be forced to circulate the room and gather students with other variables that could potentially help to form one of the equations. For example if a student is given an index card that has an x on it he can either try to find students to help him make the 1st or the 3rd equation. If he wanted to make the 1st equation (x=xi+vit+1/2at2) he would have to find someone with an xi, vi, a ½, 2 t’s, and a few addition signs. This exercise will help the students to better understand and remember the 3 kinematics equations and can also be an entertaining experience for them. I will not allow them to see their notes to remember the equations they will have to come up with them on their own as a class. I will have passed out just enough symbols to form the 3- kinematics equations that I taught them in lesson 4.
EXPLORE A:
After this activity I will inform the students that there will be another “Student Challenge” for the day. This lesson will be incredibly similar to the problem session that was held during lesson 2 (1D Kinematics review). I will instruct the students to split up into groups of 2-4 and I will explain the rules of the game that they will play.
There will be 15 challenges that will be spread out in envelopes at different stations around the room. Students will have 45-60 minutes to complete as many challenges as they can and record all of their work and their answers on notepaper. Each “challenge” will be color coded in order to designate the difficulty of the task (Black=difficult, blue= intermediate, green=easy) (Yes! I do love to ski!). The more difficult problems are worth more points because they are more in depth and will take more time to complete (Black = 5 points, Blue=3 points, Green=1pt).
There will be more “challenges” than there are groups so once a group finishes with a problem they can move to an open station and won’t have to wait for all of the other students to finish with the problem that they are working on. I would also not mind if there are more than one group working on the same problem at once.
Note: I will keep small whiteboards at each station so that groups can jot down their thoughts as they try to work through the problems.
At the end of the problem session I will collect everyone’s work and I will assign a grade to each student based on the neatness of his or her work. I will be looking to see that they follow the problem solving steps and will not necessarily be looking to see if they got the correct answer. However, for the purpose of the competition I will give groups points for correct answers to problems and I will tally everyone’s points to see which group won the day’s student challenge. Points will not be awarded to groups for partially correct answers so it is more beneficial for students to be thorough and detailed during this activity than for them to speed through all of the problems. The members of the winning group will receive extra credit points towards the final assessment of the unit.
If students seem to be struggling with the problems that I have assigned to them I will walk around the classroom and offer them assistance.
“What is the question really asking?”
“Go back and look at the example that we did together as a class last week and try to follow that same procedure here”
“What variables were given and which are we trying to find”
“Draw a picture to help yourself visualize the situation”
“This particular equation might be useful to you in this situation”
“If we know the initial velocity and the angle then we know the initial velocity in both the x and the y direction. How?”
Within each envelope (for each problem) I will have three other small sheets of paper. On first sheet I will have drawn a picture of the situation for the students. On the second sheet I will have identified the given variables for the students. And on the third sheet I will have identified the appropriate equations for the problem. I will encourage students not to look at these “hints” but if they need to they can and this should help them to more successfully complete the problems. In the beginning they may need to look at the hints but eventually they should become proficient enough with the problem solving strategies that they can do this on their own.
For the students that may find some of these problems too easy I have included several very challenging problems that they can try to complete. This is a good way for them to challenge themselves and test their knowledge of projectile motion. If this is still not enough for them I will give them a practice exam (from the Barron’s AP book) that they can work on.
SUMMARIZE/SHARE A:
After the students have been given ample time to complete many of the problems I will instruct them that they must all turn in their work so that I can grade it and determine which group won. The students will return to their desks and If there is time remaining in the class period I will ask them if they would like to review any of the problems that they had worked on. We will do some of these problems together as a class and I will have some groups assist me in doing the problems on the whiteboard at the front of the classroom.
Launch A:
During the launch phase of this lesson I will informally assess the students on their ability to remember and recreate the 3 kinematics equations that are central to solving projectile motion problems. I will observe them as they work together to form the equations out of the index cards that I will have passed out to each of them.
Explore A:
During the explore phase of this lesson I will walk around the classroom and will informally assess the students on their ability to work cooperatively in groups and on their problem solving skills. I will question the groups while they are working together and try to gauge each students understanding of the material. I will determine if more time may need to be spent on the concepts that had been recently presented to the students. The end of the unit is nearing and I want to make sure that all students will be ready for final assessment that will take place the following week. I may need to ask some students to stay after school to practice more problems with me.
Summarize A:
The students will pass in their work from the day to me at the end of the problem session. I will grade each individual’s work for completion and will specifically look to make sure that they are explicitly following the steps to problem solving. I want to make sure that they continue to use these steps even when the problems seem overly simple to them. Once they encounter more complicated problems they will need to outline and detail each step in order to be successful. The students will receive a quiz grade for their work and I will not dock points for incorrect answers or if they aren’t able to complete all of the problems.
This section to be completed only if lesson plan is implemented.